Remarks on continuous images of Radon-Nikod´ ym compacta
M. Fabian†, M. Heisler, E. Matouˇskov´a††
Abstract. A family of compact spaces containing continuous images of Radon-Nikod´ym compacta is introduced and studied. A family of Banach spaces containing subspaces of Asplund generated (i.e., GSG) spaces is introduced and studied. Further, for a continu- ous image of a Radon-Nikod´ym compactKwe prove: IfKis totally disconnected, then it is Radon-Nikod´ym compact. IfKis adequate, then it is even Eberlein compact.
Keywords: Asplund generated space, continuous image of Radon-Nikod´ym compact, totally disconnected compact, adequate compact, Eberlein compact
Classification: 46B22
Introduction
A Banach spaceX is calledAsplund if every subspace of it has separable dual.
Xis calledAsplund generated (orGSG) if it contains a linear continuous image of an Asplund space as a dense set. Thus, every weakly compactly generated space is Asplund generated.
All topological spaces in this note are assumed to be Hausdorff. A compact space is called Radon-Nikod´ym if it can be found, up to a homeomorphism, in the dual to an Asplund space, endowed with the weak∗ topology. Note that every Eberlein compact is Radon-Nikod´ym. We recall
Theorem 0 ([St1], [F, Theorem 1.5.4]). For a compact space K the following assertions are equivalent:
(i) K is a Radon-Nikod´ym compact;
(ii) C(K)is an Asplund generated space;
(iii) the dual unit ball BC(K)∗, w∗
endowed with the weak∗ topology is a Radon-Nikod´ym compact.
Corollary 1([F, Theorem 1.5.5]). For a compact spaceKthe following assertions are equivalent:
(i) K is a continuous image of a Radon-Nikod´ym compact;
(ii) C(K)is a subspace of an Asplund generated space;
(iii) BC(K)∗, w∗
is a continuous image of a Radon-Nikod´ym compact.
† Supported by grants AV 101-95-02, AV 101-97-02, and GA ˇCR 201-94-0069.
†† Supported by grant GA ˇCR 201-94-0069.
Corollary 2([F, Theorem 1.5.6]). For a Banach spaceZ the following assertions are equivalent:
(i) Z is a subspace of an Asplund generated space;
(ii) the dual unit ball BZ∗, w∗
is a continuous image of a Radon-Nikod´ym compact;
(iii) C((BZ∗, w∗))is a subspace of an Asplund generated space.
A main open question raised by Namioka [N1], [N2], but going back to Grothen- dieck’s memoir [Gr], sounds as:
(∗) Is a continuous image of a Radon-Nikod´ym compact a Radon-Nikod´ym compact?
A related question for Banach spaces has a negative answer: There exists an Asplund generated space and its subspace which is not Asplund generated.
Indeed, Stegall [St1] observed that Rosenthal’s counterexample [Ro] of a weakly compactly generated space (L1 on a “big” measure space) and its non weakly compactly generated subspace fits the job. For another example (now of the form C(K)) see [A1]. However, according to [BRW], the dual unit ball of such subspaces is Eberlein, hence Radon-Nikod´ym compact.
Note also that if (∗) has a positive answer then (∗∗) below has a positive answer and vice versa.
(∗∗) If Z is a subspace ofX and(BX∗, w∗)is a Radon-Nikod´ym compact, is such the compact(BZ∗, w∗)?
To see this, letϕbe a continuous mapping of a Radon-Nikod´ym compactLonto a compact K. Then the assignment f 7→ f ◦ϕ maps the Banach space C(K) onto a closed subspace ofC(L) isometrically. We observe that (BC(L)∗, w∗) is a Radon-Nikod´ym compact by Theorem 0. Now assume that (∗∗) has a positive answer. Then (BC(K)∗, w∗) is a Radon-Nikod´ym compact and hence so isK.
Note that if (BZ∗, w∗) is a Radon-Nikod´ym compact, then Z may not be Asplund generated, see [F, Theorem 1.6.3].
The aim of this note is twofold. First we define and study a class of com- pacta which is, at least formally, larger than that of continuous images of Radon- Nikod´ym compacta. We call them countably lower fragmentable compacta. Par- alelly we do the same for a Banach space counterpart of such compacta. Namely, we consider Banach spaces whose dual unit ball with the weak∗topology is count- ably lower fragmentable. This class extends, at least formally, the class of sub- spaces of Asplund generated spaces.
The second part studies our concepts in the framework of totally disconnected compacta. We prove, in a different way, a result of Argyros thata totally discon- nected compact, which is a continuous image of a Radon-Nikod´ym compact, is Radon-Nikod´ym compact([A2]). We also get that,an adequate compact, which is a continuous image of a Radon-Nikod´ym compact is even Eberlein compact.
We hope that this note will bring a bit of light to the open questions mentioned above.
Countably lower fragmentable compacta
LetX be a Banach space,H a set in X∗,A a bounded set inX and ∆>0.
We say that
(i) H is (A,∆)-fragmentable if for every nonempty set M ⊂ H there is a weak∗ open setG⊂X∗ such thatM ∩G6=∅and
diamA(M ∩G) := sup{hx∗1−x∗2, xi:x∗1, x∗2 ∈M∩G, x∈A} ≤∆;
(ii) H is (A,∆)-dentable if for every nonempty setM ⊂H there are x∈ X andα >0 such that diamAS(M, x, α)≤∆, where
S(M, x, α) ={x∗∈M : hx∗, xi>suphM, xi −α};
(iii) H is (A,∆)-separable if there exists an at most countable setC⊂H such that for everyx∗ ∈H there isy∗∈C such that sup{|hx∗−y∗, xi|: x∈ A} ≤∆;
(iv) the dualX∗iscountably weak∗ dentable if there exist bounded setsAn,p, n, p ∈ N, inX such that S∞
n=1An,p =X for every p∈ Nand the dual unit ballBX∗ is (An,p,1p)-dentable for everyn,p∈N. (Note that in [H1]
such anX is called a countably dentable space.)
LetKbe a compact space,Aa bounded set inC(K), and ∆>0. We say that (i) K is (A,∆)-fragmentable if for every nonempty setM ⊂ K there is an
open setG⊂K such thatM ∩G6=∅ and
diamA(M∩G) := sup{f(k1)−f(k2) : k1, k2∈M ∩G, f∈A} ≤∆;
(ii) Kis (A,∆)-separable if there exists an at most countable setC⊂Ksuch that for everyk ∈K there isk′ ∈C such that sup{|f(k)−f(k′)| : f ∈ A} ≤∆;
(iii) K iscountably lower fragmentable if there exist bounded setsAn,p,n, p∈ N, inC(K) such thatS∞
n=1An,p=C(K) andKis (An,p,1p)-fragmentable for everyp∈N.
Propositions below relate the above concepts mutually and to the known no- tions of Radon-Nikod´ym compacta and Asplund generated spaces.
Proposition 1. LetK be a compact space, A a bounded subset of C(K) and
∆>0. Letκ:K→C(K)∗ be the canonical mapping sendingk∈Kto the point massδk. Then:
(i) Kis(A,∆)-fragmentable if and only if (κ(K), w∗)is(A,∆)-fragmentable if and only if κ(K)is(A,∆)-dentable;
(ii) K is(A,∆)-separable if and only if κ(K)is(A,∆)-separable;
(iii) if C(K)∗ is countably weak∗ dentable, then K is countably lower frag- mentable.
Proof: Everything is trivial but the fact that (A,∆)-fragmentability ofKimplies (A,∆)-dentability ofκ(K). So assume thatKis (A,∆)-fragmentable and consider
∅ 6= κ(M) ⊂ κ(K). We find an open set G ⊂ K such that M ∩G 6= ∅ and diamA(M∩G)≤∆. Pickk∈M∩Gand findf ∈C(K) such thatf(k) = 1 and f(k′) = 0 for allk′ ∈K\G. Then surely S:=S(κ(M), f,1/2)⊂κ(M)∩κ(G) = κ(M ∩G) and so diamAS≤diamA(M∩G)≤∆.
Proposition 2. (i) If X is an Asplund space, then X∗ is countably weak∗ dentable.
(ii) If Z is a subspace of a Banach space X and X∗ is countably weak∗ dentable, then so isZ∗.
(iii) If T :X →Y is a linear continuous mapping, withT X =Y, andX∗ is countably weak∗ dentable, then so isY∗.
(iv) If Z is a subspace of an Asplund generated space, then Z∗ is countably weak∗ dentable.
(v) If Z∗ is countably weak∗ dentable, thenZ is a weak Asplund space, that is, that every continuous convex function on Z is Gˆateaux differentiable at the points of a denseGδ set.
Proof: (i) the sets An,p =nBX, n, p ∈ N, fit the job. (ii), (iii), and (v) are proved in [H1]. (iv) follows by putting together (i), (ii) and (iii).
Proposition 3. Continuous images of Radon-Nikod´ym compacta are countably lower fragmentable.
Proof: Put together Corollary 1, Proposition 2(iv), and Proposition 1(iii).
We do not know if the converse to (iv) in Proposition 2 holds. Likewise, it is unclear if Proposition 3 can be reversed. Using Corollary 1, Theorem 1 and Theorem 2, we can check that these two questions are equivalent.
The effort below is devoted to proving a converse to the implication (iii) in Proposition 1. Actually, we prove a complete analogue of Corollary 1:
Theorem 1. For a compact spaceK the following assertions are equivalent:
(i) K is a countably lower fragmentable compact;
(ii) the dualC(K)∗ is countably weak∗ dentable;
(iii) the dual unit ball (BC(K)∗, w∗) is a countably lower fragmentable com- pact.
The proof is a compilation of several lemmas listed (and proved) below.
Lemma 1. Let K be a compact space, A ⊂ C(K) a bounded set and ∆ > 0.
Then K is (A,∆)-fragmentable if and only if for every at most countable set A0⊂Athe spaceK is(A0,∆)-separable.
Proof: Throughout the whole proof we use techniques known from the theory of Asplund spaces, see [Ph]. Assume that K is (A,∆)-fragmentable and take
a countable setA0⊂A. LetZbe the closed linear span ofA0; soZ is a separable Banach space. Fork∈Kdefine φ(k)(z) =z(k),z∈Z. Thenφ(k)∈Z∗ and the mappingφ: K→(Z∗, w∗) is continuous. Hence (φ(K), w∗) is a compact space.
We claim that φ(K) is (A0,∆)-fragmentable. Take ∅ 6= M ⊂ φ(K). Using Zorn’s lemma, we find a closed setN ⊂K, minimal with respect to the inclusion, such that φ(N) is equal to the weak∗ closure M ∗ of M. Since K is (A0,∆)- fragmentable, there is an open setU⊂K such thatN∩U 6=∅ and diamA0(N∩ U)≤∆. The set N\U is closed soφ(K)\φ(N\U) is an open set in (φ(K), w∗).
We find a weak∗ open set G⊂Z∗ such thatφ(K)\φ(N\U) =G∩φ(K). Also M∗∩G=M ∗\φ(N\U) and this set is nonempty because of the minimality of N and sinceN∩U 6=∅. Further we observe thatM ∩G⊂φ(N∩U). Therefore
diamA0(M ∩G) = sup{hz1∗−z2∗, zi: zi∗∈M ∩G, z∈A0}
≤sup{hz1∗−z2∗, zi: zi∗∈φ(N∩U), z∈A0}
= sup{hφ(k1)−φ(k2), zi: ki ∈N∩U, z∈A0}
= sup{z(k1)−z(k2) : ki∈N∩U, z∈A0}
= diamA0(N∩U)≤∆.
The claim is thus proved.
Now assume thatKis not (A0,∆)-separable. Then there exists an uncountable setC⊂K such that
sup{|f(k)−f(k′)|: f ∈A0}>∆ whenever k, k′∈C and k6=k′. Then
sup{|hφ(k)−φ(k′), fi|: f ∈A0}>∆ whenever k, k′∈C, and k6=k′. SinceZ is a separable Banach space, there is a sequence{zn: n∈N}contained in and dense in the unit ball of Z. Define a mapping ψ : (Z∗, w∗) → RN by the formula ψ(z∗)(n) =hz∗, zni, z∗ ∈Z∗, n∈ N; it is injective and continuous.
It then follows that ψ(φ(K)) is a metrizable compact. Hence (φ(K), w∗) is a metrizable compact andL := (φ(C), w∗) is a metrizable separable space. Thus, the topology onLhas a countable basis, sayB. PutB0={U ∈ B: L∩U is at most countable}. Then the setL∩ SB0
is at most countable and hence the set Le=L\ S
B0
is nonempty. We shall check thatLehas no isolated point. So take anyz∗∈L. Ife z∗∈U ∈ B, thenU /∈ B0 and hence the setL∩U is uncountable and so is the setLe∩U. Thereforez∗ is not an isolated point ofL.e
Now we apply the claim to the (nonempty) set M := Le ⊂ φ(K). We get a weak∗ open setG⊂Z∗such thatLe∩G6=∅and diamA0(eL∩G)≤∆. According to the property of the setC, we then conclude that Le∩G is a singleton. This means that Le∩Gconsists of an isolated point of L. Howevere Le does not have
isolated points. This is a contradiction and therefore K is (A0,∆)-separable.
Second, assume K is not (A,∆)-fragmentable. Fors∈ {0,1} ∪ {0,1}2∪. . . we shall construct nonempty open sets Gs ⊂ K, functions fs ∈ A, and numbers
∆s>∆ as follows: PutG0 =G1 =K. Assume that for somes= (s1, . . . , sn) we already have nonempty open sets Gs ⊂K with diamAGs>∆. We findfs ∈A and k0, k1 ∈ Gs such that fs(k1)−fs(k0) > ∆. Then we find ∆s satisfying fs(k1)−fs(k0)>∆s>∆. Takea >0 so thatfs(k1)> a > fs(k0) + ∆s. Then put
Ges,0={k∈Gs: fs(k)< a−∆s}, Ges,1={k∈Gs: fs(k)> a}.
Thuskj∈Ges,j,j = 0,1. Find open setsGs,j such thatkj∈Gs,j ⊂Gs,j⊂Ges,j, j= 0,1. This finishes the induction step. Now putA0={fs: s∈ {0,1}∪{0,1}2∪ . . .}; this is a countable subset ofA. For everyσ= (s1, s2, . . .)∈ {0,1}Nchoose kσ∈T
n∈NGs1,s2,...,sn; such a kσ exists. Observe that
sup{|f(kσ)−f(kσ′)|: f ∈A0}>∆ whenever σ, σ′∈ {0,1}N, and σ6=σ′.
Therefore the spaceK is not (A0,∆)-separable.
Lemma 2. Let K be a compact space, A ⊂ C(K) a bounded countable set and ∆ > 0. If K is (A,∆)-separable, then the unit ball BC(K)∗ in C(K)∗ is (A,2∆)-separable.
Proof: We shall immitate an argument due to W.B. Moors, see the proof of [F, Lemma 1.5.3]. Let{kn : n∈N} be a sequence which is (A,∆)-dense in K, i.e.
for everyk∈K there existsn∈Nsuch that sup{|f(k)−f(kn)|: f ∈A} ≤∆.
Denote
H={ν∈C(K)∗ : sup|hν, Ai| ≤∆},
and Kn=
k∈K: δk∈ {δk1, . . . , δkn}+H , n∈N.
(Here δk ∈ C(K)∗ means the point mass measure at k ∈ K.) Clearly H is convex symmetric and weak∗ closed, Kn is closed, andK=S∞
n=1Kn. Take any µ∈BC(K)∗. (We use here and below F. Riesz’ representation theorem.) We find n∈Nso that|µ|(K\Kn)<∆/(2c) wherec= sup{kfk: f ∈A}. (|µ| means the total variation ofµ.) Define
ν(M) =µ(M ∩Kn), M ⊂K Borel.
Thenν ∈BC(K)∗ andkµ−νk ≤ |µ|(K\Kn)<∆/(2c); soµ∈ν+12H. Now we observe, using the separation theorem, thatνbelongs to the weak∗ closed convex hull of{±δk: k∈Kn}. Thus, denoting S= co{±δk1,±δk2, . . .}, we have
µ∈ν+12H ⊂co∗{±δk: k∈Kn}+12H
⊂co{±δk1, . . . ,±δkn}+H+12H⊂S+32H.
ThereforeBC(K)∗ ⊂S+32H. Now we observe that S is a norm separable set, henceS is also (A,∆/2)-separable. ThusBC(K)∗ is (A,2∆)-separable.
Lemma 3. LetX be a Banach space,Aa bounded set inX,∆>0and assume thatBX∗ is(A,∆)-fragmentable. ThenBX∗ is(A,2∆)-dentable.
Proof: We just copy the proof of (iv)⇒(i) in [NP, Lemma 3].
Lemma 4. LetLbe a compact space,X a linear subset ofC(L)which separates the points of L, and ∆ > 0. Suppose that there exist bounded sets An ⊂ X, n∈N, such thatS∞
n=1An =X and L is(An,∆)-fragmentable for eachn∈N. Then there exist bounded setsF1 ⊂F2⊂ · · · ⊂C(L)such thatS∞
n=1Fn=C(L), andLis(Fn,2∆)-fragmentable for each n∈N.
Proof: Denote by 1 the function on L which is identically equal to one. We observe thatL is (A,∆)-fragmentable where A is the convex symmetric hull of the setAn∪{n·1}. Therefore, by considering the linear span of{1} ∪Xinstead of X and the convex symmetric hull ofAn∪ {n·1}instead ofAnfor eachn∈N, we may, and do assume thatX contains the constants and the setsAnare bounded, convex, and symmetric.
Put E1 = A1. If En was already defined for some n ∈ N, let En+1 be the convex symmetric hull of the set
{f1∨f2: f1, f2 ∈An+1∪En},
wheref∨gdenotes the pointwise maximum of the functionsf andg. ClearlyLis (E1,∆)-fragmentable. SupposeLis (En,∆)-fragmentable for some n∈N. Then L is also (An+1∪En,∆)-fragmentable. Take f, g ∈ En∪An+1 and l1, l2 ∈ L.
Then
(f∨g)(l1)−(f∨g)(l2)≤max{f(l1)−f(l2), g(l1)−g(l2)} ≤diamEn∪An+1{l1, l2}.
HenceLis also (B,∆)-fragmentable whereB={f∨g: f, g∈En∪An+1}, and finally,Lis (En+1,∆)-fragmentable.
Clearly,E1 ⊂E2 ⊂. . ., and the setY =S∞
n=1Enis closed under the operation of taking pointwise maximum and minimum of two functions, separates the points of L, and contains the constant functions. We shall show that Y is a linear subset ofC(L). Then, by the Stone-Weierstrass theorem (see the proof of [DS, Theorem IV.6.16]),Y is dense inC(L). Therefore the sets
Fn=En+∆2BC(L), n∈N, have the required properties.
Since each of the sets E1 ⊂ E2 ⊂ . . . is convex and symmetric, in order to show that Y is linear it is enough to find for each n ∈ N, each f ∈ En, and eachc >0 somei ∈N so that cf ∈Ei. We shall show this by induction on n.
If f ∈E1 (= A1) andc >0 then cf ∈X and hence there exists i ∈N so that cf∈Ai⊂Ei. Suppose that the statement holds for somen∈N, and letf ∈En+1
andc >0 be given. Then there existm∈N, f1, f2, . . . , f2m∈An+1∪En, and λ1, . . . , λm∈Rsuch thatPm
j=1|λj| ≤1 and f =
Xm j=1
λj(f2j−1∨f2j).
For each j ∈ {1, . . . ,2m} we find ij ∈ N so that cfj ∈ Eij, and put i = 1 + max{i1, i2, . . . , i2m}. Then
cf = Xm j=1
λj((cf2j−1)∨(cf2j))∈Ei.
Proof of Theorem 1: : (i)⇒(ii). Assume that K is a countably lower frag- mentable compact. Let An,p be the sets from the definition of countable lower fragmentability. Fix anyn, p∈N. We know thatKis (An,p,p1)-fragmentable. By Lemma 1, for every countable set A0 ⊂An,p, the spaceK is (A0,1p)-separable.
Then, by Lemma 2, BC(K)∗ is (A0,2p)-separable for every such A0. Again, by Lemma 1, BC(K)∗ is (An,p,2p)-fragmentable. (Here, in order to be able to use Lemma 1, we consider A0 and An,p as if they were subsets of C((BC(K)∗, w∗))
— this can be ensured by a canonical embedding.) Finally, by Lemma 3,BC(K)∗ is (An,p,4p)-dentable. This means nothing else than that (ii) holds.
(ii)⇒(iii). Let K satisfy (ii) in our theorem. We observe that the Banach space C(K) embeds isometrically as a closed subspace into C((BC(K)∗, w∗)) and this subspace separates the points of BC(K)∗. By (ii), there exist sets An,p ⊂ C(K), n, p ∈ N, so that S∞
n=1An,p = C(K) for each p ∈ N and that BC(K)∗ is (An,p,1p)-fragmentable for each n, p ∈ N. Fix any p ∈ N and ap- ply Lemma 4 for L := (BC(K)∗, w∗), X := C(K) ⊂ C((BC(K)∗, w∗))
, and An := An,p ⊂ C((BC(K)∗, w∗))
, n ∈ N. In this way, we get bounded sets Fn,p ⊂ C((BC(K)∗, w∗)), n, p ∈ N, so that S∞
n=1Fn,p = C((BC(K)∗, w∗)) for each p ∈ N, and the compact space (BC(K)∗, w∗) is (Fn,p,2p)-fragmentable for eachn, p∈N.
(iii)⇒(i). Let An,p ⊂ C((BC(K)∗, w∗)), n, p ∈ N, do the job in (iii). Then the setsA′n,p =
f|K : f ∈An,p do the job for (i). (We assumed that K is a
subspace of (BC(K)∗, w∗).)
Theorem 2. For a Banach spaceX the following assertions are equivalent:
(i) the dual spaceX∗ is countably weak∗ dentable;
(ii) the compact space(BX∗, w∗)is countably lower fragmentable;
(iii) the dualC((BX∗, w∗))∗ is countably weak∗dentable.
Proof: (i)⇒(ii) follows from Lemma 4. (ii)⇒(iii) follows from (i)⇒(ii) in Theo- rem 1. (iii)⇒(i) follows from Proposition 2(ii).
Countably lower fragmentable compacta which are totally disconnected
A topological space is called totally disconnected if its topology has a basis consisting from sets which are both open and closed. Thus, a compact space is totally disconnected if and only if it is homeomorphic to a closed subspace of the product {0,1}Γ for some set Γ. For A ⊂ Γ we define χA : Γ → {0,1}
as χA(γ) = 1 if γ ∈ A and χA(γ) = 0 if γ ∈ Γ\A. If k ∈ {0,1}Γ, we put suppk ={γ ∈ Γ : k(γ) = 1}. A compact space K is called adequate if it is a closed subspace of{0,1}Γ for some set Γ,
(i) χ{γ}∈Kfor everyγ∈Γ, and
(ii) ifB⊂A⊂Γ andχA∈K, thenχB∈K.
Theorem 3. For a totally disconnected compact spaceKthe following assertions are equivalent:
(i) K is a Radon-Nikod´ym compact(i.e.,C(K)is Asplund generated);
(ii) K is a continuous image of a Radon-Nikod´ym compact (i.e., C(K)is a subspace of an Asplund generated space);
(iii) K is a countably lower fragmentable compact (i.e., C(K)∗ is countably weak∗ dentable);
(iv) if ϕis a homeomorphism ofK into{0,1}Γ, then there exist setsΓn⊂Γ, n∈N, with S∞
n=1Γn= Γ, such that for everyn∈Nand every∅ 6=M ⊂ Kthere are an open setG⊂M andΓ′ ⊂Γnsuch thatsuppϕ(k)∩Γn= Γ′ wheneverk∈M ∩G.
If K is an adequate compact, then the above conditions are equivalent with:
(v) K is an Eberlein compact(i.e.,C(K)is weakly compactly generated).
Proof: (i)⇒(ii) is trivial. (ii)⇒(iii) is Proposition 3.
(iii)⇒(iv) Let An,p, n, p ∈ N, be the sets guaranteeing the countable lower fragmentability of K and let ϕ : K → {0,1}Γ be a continuous injection. For γ∈Γ putπγ(k) =ϕ(k)(γ),k∈K; thusπγ∈C(K). Define
Γn={γ∈Γ : πγ∈An,2}, n∈N. ThenS∞
n=1Γn = Γ. Now take∅ 6=M ⊂K and fixn∈N. We find an open set G⊂K such that M ∩G 6=∅ and diamAn,2(M ∩G)≤ 12. Take k, k′ ∈ M ∩G.
Then for everyγ∈Γn we haveπγ∈An,2 and so
|πγ(k)−πγ(k′)| ≤ 1 2.
This means that suppϕ(k)∩Γn= suppϕ(k′)∩Γn and so (iv) is satisfied.
(iv)⇒(i) Letϕ:K→ {0,1}Γ be a continuous injection and let Γn, n∈N, be found in (iv). Put
An={πγ: γ∈Γn}, n∈N, and define
ρ(k, k′) = supn
|f(k)−f(k′)|: f ∈A1∪12A2∪13A3∪ · · ·o
, k, k′∈K.
Clearly,ρis a lower semicontinuous metric onK. We shall show thatρfragments K, that is, that every nonempty subset ofKcontains a nonempty relatively open subset whoseρ-diameter is less than an a priori given arbitrary positive number.
Then, by [N2], we can conclude that K is a Radon-Nikod´ym compact. So let
∅ 6= M ⊂ K and ǫ > 0 be given. Take n∈ N so that 1/n < ǫ. By (iv), there is an open setG1 ⊂K, with M ∩G1 6=∅, and such that the set suppϕ(k)∩Γ1
does not depend uponk ∈ M ∩G1. Then, again by (iv), there is an open set G2⊂K, with (M∩G1)∩G26=∅, and such that the set suppϕ(k)∩Γ2 does not depend uponk∈(M∩G1)∩G2. Continuing in this way, we finally find an open setGn⊂K, withM∩G1∩ · · · ∩Gn6=∅, and such that suppϕ(k)∩Γndoes not depend uponk∈M ∩G1∩ · · · ∩Gn. PutG=G1∩ · · · ∩Gn. ThenM ∩G6=∅ and the set suppϕ(k)∩(Γ1∪ · · · ∪Γn) does not depend uponk∈M∩G. Thus, fork, k′∈M ∩Gwe have
ρ(k, k′) = supn
|f(k)−f(k′)|: f ∈n+11 An+1∪n+21 An+2∪ · · ·o
≤ 1 n+ 1 < ǫ.
This means thatK is fragmented byρand (i) is proved.
Finally, assume thatK is an adequate compact. We find a set Γ so thatK is a subspace of{0,1}Γ. Assume thatK satisfies (iv). Let Γn, n∈N, be the sets from (iv). By replacing the set Γ2 by Γ2\Γ1, the set Γ3 by Γ3\(Γ1∪Γ2) and so on, we may, and do assume that Γi∩Γj =∅wheneveri6=j. We claim that for everyk∈K and every n∈Nthe set suppk∩Γnis finite. Then the assignment k 7→ {n1k(γ) : if γ ∈ Γn, n ∈ N} sends K into (c0(Γ), w) continuously and injectively and henceK is an Eberlein compact.
Assume that the claim is false. Then there exist n∈N andk∈K such that suppk∩Γn is an infinite set. LetA⊂Γ be such thatk=χA; thenA∩Γnis an infinite set. Denote
M ={χB: B⊂A∩Γn}.
Since K is adequate compact, M is a (nonempty) subset of K. By (iv), there exists an open setG⊂K such thatM ∩G6=∅ and (B=)B∩Γn= const. for everyχB∈M∩G. HenceM∩Gis a singleton. However, this contradicts to the definition of the setM and to the definition of the topology on{0,1}Γ. In [T], Talagrand constructed an adequate compactKwhich is not Eberlein. In [OSV], it is shown that thisK is not a Radon-Nikod´ym compact. Stegall showed in [St2] that thisK is not a continuous image of a Radon-Nikod´ym compact, see also [F, Theorem 8.3.6]. From Theorem 3 we get the same fact in an easier way.
HenceC(K) is not a subspace of an Asplund generated space.
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Mathematical Institute, Czech Academy of Sciences, ˇZitn´a 25, 115 67 Prague 1, Czech Republic
(Received December 30, 1996,revised June 17, 1997)
